Minimal Polynomials of Linear Recurrence Sequences#
AUTHORS:
William Stein
- sage.matrix.berlekamp_massey.berlekamp_massey(a)#
Use the Berlekamp-Massey algorithm to find the minimal polynomial of a linear recurrence sequence \(a\).
The minimal polynomial of a linear recurrence \(\{a_r\}\) is by definition the unique monic polynomial \(g\), such that if \(\{a_r\}\) satisfies a linear recurrence \(a_{j+k} + b_{j-1} a_{j-1+k} + \cdots + b_0 a_k=0\) (for all \(k\geq 0\)), then \(g\) divides the polynomial \(x^j + \sum_{i=0}^{j-1} b_i x^i\).
INPUT:
a– a list of even length of elements of a field (or domain)
OUTPUT:
the minimal polynomial of the sequence, as a polynomial over the field in which the entries of \(a\) live
Warning
The result is only guaranteed to be correct on the full sequence if there exists a linear recurrence of length less than half the length of \(a\).
EXAMPLES:
sage: from sage.matrix.berlekamp_massey import berlekamp_massey sage: berlekamp_massey([1,2,1,2,1,2]) x^2 - 1 sage: berlekamp_massey([GF(7)(1), 19, 1, 19]) x^2 + 6 sage: berlekamp_massey([2,2,1,2,1,191,393,132]) x^4 - 36727/11711*x^3 + 34213/5019*x^2 + 7024942/35133*x - 335813/1673 sage: berlekamp_massey(prime_range(2, 38)) # needs sage.libs.pari x^6 - 14/9*x^5 - 7/9*x^4 + 157/54*x^3 - 25/27*x^2 - 73/18*x + 37/9